Introduction
For the moment this vignette shows only minimal examples on how to use the moin package. These are from textbooks presenting methods on singly- and doubly-constrained interaction models. The examples correspond to those from the help document to the functions. However, here also the expected results are shown.
Singly-constrained model – A location model.
\[ flows_{ij} = O_i \frac{W_j^\alpha c_{ij}^{-\beta}}{\sum_j W_j^\alpha c_{ij}^{-\beta}} \]
where \(O_i\) are the sources of flows and \(W_j\) is the attractiveness of the target. It can be weighted by \(\alpha\). \(c_{ij}\) are the travel costs that can be weighted with \(\beta\).
Load the package
library(moin)Create some data. These stem from Wilson and Kirkby (1980), p. 100f.
Piis the population of zone ieiis the mean expenditure on shopping goods per head in zone iWjis the attractiveness of shops in zone jcijis the cost of travel from i to j
ei <- c(2,1,1)
Pi <- c(50, 1000, 500)
Wj <- c(10, 100, 20)
cij <- matrix(data = c(1, 5, 5,
5, 2.585, 5,
5, 5, 2),
nr = 3,
nc = 3
)Apply the model
sc(Oi = ei * Pi, Wj = Wj, cij = cij, detfun = "power")## $flows
## [,1] [,2] [,3]
## [1,] 29.41176 58.82353 11.76471
## [2,] 44.75803 865.72591 89.51606
## [3,] 31.25000 312.50000 156.25000
##
## $Dj
## [1] 105.4198 1237.0494 257.5308Here are the original data. \(S_{11}\) to \(S_{33}\) correspond to the flows as returned by sc function. Dj corresponds to \(\sum_i S_{ij}\)
What is important here? The main characteristics is that the flow totals are constrained by \(e_i * P_i\). Due to this it can act as a location model and predict the spatial distribution of shopping sales (these are the Dj values).
Doubly-constrained model
\[ T_{ij} = A_i O_i B_j D_j c_{ij}^{-\beta} \]
with constraints
\[ \sum_j^n T_{ij} = O_i \]
and
\[ \sum_i^n T_{ij} = D_j \]
Therefore, two balancing scalars are necessary to ensure that the origin (\(A_i\)) and destination (\(B_j\)) constraints are satisfied.
\[ A_i = (\sum_j^n B_j D_j c_{ji}^{-\beta})^{-1} \]
and
\[ B_j = (\sum_i^n A_i O_i c_{ji}^{-\beta})^{-1} \]
Create some data. These stem from Thomas and Huggett (1980), p. 150. It is a journey-to-work example (see figure)
Oi refers to workers in zone i and Dj to jobs in j.
Oi <- c(4,6,2)
Dj <- c(3,8,1)
cij <- matrix(data = c(1,2,2,
2,1,2,
2,2,1
),
nr = 3,
nc = 3
)
beta <- 1Apply the model
tmp <- dc(Oi = Oi, Dj = Dj, cij = cij, iterations = 5)
str(tmp)## List of 10
## $ iteration: num 4
## $ beta : num 1
## $ Oi :'data.frame': 3 obs. of 2 variables:
## ..$ Target: num [1:3] 4 6 2
## ..$ sj : num [1:3] 4 6 2
## $ Dj :'data.frame': 3 obs. of 2 variables:
## ..$ Target: num [1:3] 3 8 1
## ..$ si : num [1:3] 3 8 1
## $ Ratio :'data.frame': 3 obs. of 2 variables:
## ..$ rj: num [1:3] 1 1 1
## ..$ ri: num [1:3] 0.75 1.33 0.5
## $ error : num 0.000348
## $ Ai :'data.frame': 4 obs. of 3 variables:
## ..$ X1: num [1:4] 1 0.133 0.133 0.133
## ..$ X2: num [1:4] 1 0.1 0.101 0.101
## ..$ X3: num [1:4] 1 0.154 0.152 0.152
## $ Bj :'data.frame': 4 obs. of 3 variables:
## ..$ X1: num [1:4] 1 1.01 1.02 1.02
## ..$ X2: num [1:4] 1 0.98 0.979 0.979
## ..$ X3: num [1:4] 1 1.14 1.15 1.15
## $ Tij : num [1:3, 1:3] 1.617 0.92 0.463 2.078 4.733 ...
## $ sumTij : num 12Now, the model output is a little longer. Oi and Dj have Target and si or sj columns. The former are the input data, the latter the calculated values.
tmp$Ai## X1 X2 X3
## 1 1.0000000 1.0000000 1.0000000
## 2 0.1333333 0.1000000 0.1538462
## 3 0.1327950 0.1006996 0.1519117
## 4 0.1327341 0.1007419 0.1518597tmp$Bj## X1 X2 X3
## 1 1.000000 1.0000000 1.000000
## 2 1.012987 0.9798995 1.143695
## 3 1.015032 0.9787616 1.147431
## 4 1.015206 0.9786849 1.147561Ai and Bj refer to the balancing factors. Since these result from simultaneous equations they need to be find by iteration. iteration states that the model ran for four iterations, although we stated five explicitly in the model call. The reason is, that the model was balanced already after the fourth iteration. Compared to the values of Thomas and Huggett (1980), p. 151 it gets obvious the the values are rather different (this might be caused by rounding effects throughout the calculations).
round(tmp$Tij, 2)## [,1] [,2] [,3]
## [1,] 1.62 2.08 0.30
## [2,] 0.92 4.73 0.35
## [3,] 0.46 1.19 0.35Tij shows the flows and these correspond to those from Thomas and Huggett (1980), p. 152
